**1. Introduction**

It is a well-established fact that a moving body reduces drag if it is elevated by a layer of air. This phenomenon is used in air-cushioned vehicles and in air hockey, in which the frictional resistance of moving objects is reduced. Skalak and Wang [1] were the pioneers of studying the three-dimensional flow that arises between a moving porous flat plate and the ground, and they later on wrote an erratum on their own paper [2]. Wang also studied elliptical porous sliders [3]. In the case of Newtonian fluids, past studies have included porous circular, long, inclined, and elliptical sliders. R. C. Bhattacharjee studied a porous slider bearing lubricated with a coupled stress (a magneto-hydrodynamic (MHD) fluid) [4]. Jimit made a comparison of the different porous structures on the performance of a magnetic fluid [5]. Prawal Sinha analyzed the thermal effects of a long porous rough slider bearing [6]. Mohmmadrayian analyzed a rough porous inclined slider bearing lubricated with a ferrofluid in consideration of slip velocity [7]. Ji Lang both theoretically and experimentally investigated the transient squeezing flow in a highly porous film [8]. Similarly, a large amount of literature is available in relation to long porous sliders (LPSs) [1,6,9–13] and circular porous sliders (CPSs) [2,14–18]. Awati investigated the lubrication of a long porous slider by using the homotopy analysis method (HAM) [10]. In a separate study, Khan studied the e ffects of Reynolds numbers by using di fferent analytical methods [11,12]. Naeem studied the influence of Reynolds numbers (R) on long [13] and circular porous sliders [18]. Ghoreishi studied the circular slider [14]. Madani investigated the circular porous slider by using HPM, and also analyzed its lift and drag [19].

All the above mentioned studies were done without a slip condition on either the immobile ground or slider. However, a slip condition is essential for super-hydrophobic planes, as it is di fficult to have a zero mean tangential velocity from where the fluid is injected when there is a slip. Furthermore, in order to minimize adhesion, the fluid could be a rarefied gas, where the compact exterior could be coated with a material, or the ground could be uneven so that an equivalent slip exists or there is a slip flow regime. Wang [16] discussed slip e ffects, but didn't consider the e ffects of a transverse magnetic field. Therefore, the goal of the current work is to examine the impact of slip and Reynolds numbers when a transverse magnetic field is a ffecting the performance of a porous slider. Through the literature survey, it is assumed that a three-dimensional flow with slip and a uniform magnetic field does not exist. Hence, the goal of the current research is to analyze the performance of porous sliders in the presence of slip and a Reynolds number with a constant magnetic field, and to assess their e ffects on the components of velocity lift and drag.

The structure of the article is as follows: In the introduction, a brief history of the problem of the porous slider and its application is presented. In the second section, the formulation of the problems are given, while in the third section the formulation of a homotopic solution is presented [20]. The fourth section deals with the convergence criteria of the HAM. Results and discussions are given in the fifth section. Finally, the conclusion is given in the sixth section, with a list of nomenclature.

As discussed above, the velocity slip condition is considered in this study. Navier introduced the slip condition for the first time as follows:

$$\mathbf{x}\_1 = H\boldsymbol{\zeta} \tag{1}$$

In Equation (1), tangential velocity *u* is proportional to the shear stress and *H* is the constant of proportionality, which is actually a slip coe fficient. In order to ignore the end e ffects, it is assumed that the gap between slider and ground is quite small as compared to the slider's lateral dimension. Both circular and long porous sliders are considered in this study.

### **2. Problem Formulation of Long and Circular Sliders**

In this study, the incompressible and steady flow of a viscous fluid between porous (long and circular) sliders and the ground is considered in the presence of a uniform magnetic field, as shown in Figure 1.

**Figure 1.** (**a**) Schematic diagram of the movement of a long porous slider (LPS). (**b**) Schematic diagram of the movement of a circular porous slider (CPS).

Length and width are quite big compared to height *d*. The slider moves with the velocity components and is elevated because of the injection of fluid from below with a magnetic field applied externally. In order to avoid the induced magnetic field formed by the movement of the fluid, it is assumed that the magnetic Reynolds number is not very big. Furthermore, the induced and imposed electric field are supposed to be negligible, and therefore the electromagnetic body force per unit volume simplifies F*em* = <sup>σ</sup>0(<sup>v</sup> × B) × B, where B =(0, 0, *<sup>B</sup>*0) is the magnetic field.

Under the above-stated assumptions and conditions, Navier–Stokes equations take the following form:

$$
\phi\_1 \frac{\partial \phi\_1}{\partial \mathbf{x}\_1} + \phi\_2 \frac{\partial \phi\_1}{\partial \mathbf{x}\_2} + \phi\_3 \frac{\partial \phi\_1}{\partial \mathbf{x}\_3} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_1} + \nu \left( \frac{\partial^2 \phi\_1}{\partial \mathbf{x}\_1^2} + \frac{\partial^2 \phi\_1}{\partial \mathbf{x}\_2^2} + \frac{\partial^2 \phi\_1}{\partial \mathbf{x}\_3^2} \right) - \frac{\sigma\_0}{\rho} B\_0^2 \phi\_1 \tag{2}
$$

$$
\phi\_1 \frac{\partial \phi\_2}{\partial \mathbf{x}\_1} + \phi\_2 \frac{\partial \phi\_2}{\partial \mathbf{x}\_2} + \phi\_3 \frac{\partial \phi\_2}{\partial \mathbf{x}\_3} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_2} + \nu \left( \frac{\partial^2 \phi\_2}{\partial \mathbf{x}\_1^2} + \frac{\partial^2 \phi\_2}{\partial \mathbf{x}\_2^2} + \frac{\partial^2 \phi\_2}{\partial \mathbf{x}\_3^2} \right) - \frac{\sigma\_0}{\rho} B\_0^2 \phi\_2 \tag{3}
$$

$$
\phi\_1 \frac{\partial \phi\_3}{\partial \mathbf{x}\_1} + \phi\_2 \frac{\partial \phi\_3}{\partial \mathbf{x}\_2} + \phi\_3 \frac{\partial \phi\_3}{\partial \mathbf{x}\_3} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_3} + \nu \left( \frac{\partial^2 \phi\_3}{\partial \mathbf{x}\_1^2} + \frac{\partial^2 \phi\_3}{\partial \mathbf{x}\_2^2} + \frac{\partial^2 \phi\_3}{\partial \mathbf{x}\_3^2} \right) \tag{4}
$$

Velocity components are expressed as (φ1, φ2, φ3), where ρ, *p*, and υ are density, pressure and kinematic viscosity, respectively. Law of conservation of mass is as follows:

$$\frac{\partial \phi\_1}{\partial \mathbf{x}\_1} + \frac{\partial \phi\_2}{\partial \mathbf{x}\_2} + \frac{\partial \phi\_3}{\partial \mathbf{x}\_3} = 0 \tag{5}$$

According to Naeem [13], the following transform has been used:

$$
\phi\_1 = lI\psi\_1(\boldsymbol{\varsigma}) + \frac{\mathcal{W}}{d} \mathbf{x}\_1 \psi\_3 \;/\; (\boldsymbol{\varsigma}),\\\phi\_2 = V\psi\_2(\boldsymbol{\varsigma}),\\\phi\_3 = -\mathcal{W}\psi\_3(\boldsymbol{\varsigma}).\tag{6}
$$

where ς = *x*3*d* . By adding Equation (6) into Equations (2)–(4), the following ordinary differential equations are obtained:

$$
\psi\_3^{\rm ir} = R \big( \psi\_3^{\prime} \psi\_3^{\prime \prime \prime} - \psi\_3 \psi\_3^{\prime \prime \prime \prime} \big) + M^2 \psi\_3^{\prime} \tag{7}
$$

$$
\Psi\_1^{\prime \prime} = \mathcal{R} \{ \psi\_1 \psi\_3^{\prime} - \psi\_3 \psi\_1^{\prime} \} + \mathcal{M}^2 \psi\_1 \tag{8}
$$

$$
\psi\_2^{\prime\prime\prime} = -\mathcal{R}(\psi\_1 \psi\_2^{\prime}) + M^2 \psi\_2 \tag{9}
$$

where *R* is the Reynolds number (*R* = *Wd*/υ). Boundary conditions at *x*3 = 0 and *x*3 = *d* are given in Equations (10) and (11), respectively.

$$
\phi\_1 = \mathcal{U} + H\_1 \mu \frac{\partial \phi\_1}{\partial \mathbf{x}\_3}, \quad \phi\_2 = V + H\_1 \mu \frac{\partial \phi\_2}{\partial \mathbf{x}\_3}, \quad \phi\_3 = 0 \tag{10}
$$

$$
\phi\_1 = -H\_2 \mu \frac{\partial \phi\_1}{\partial \mathbf{x}\_3}, \quad \phi\_2 = -H\_2 \mu \frac{\partial \phi\_2}{\partial \mathbf{x}\_3} = 0, \quad \phi\_3 = -W \tag{11}
$$

where *H*1, *H*2, and μ = ρυ are slip coefficients and viscosity, respectively. Equations (10) and (11) take the following form:

$$\begin{aligned} \psi\_3'(0) &= \beta\_1 \psi\_3'(0) = , \psi\_3(0) = 0, \\ \psi\_3(1) &= 1, \psi\_3'(1) = -\beta\_2 \psi\_3'(1), \\ \psi\_1(1) &= -\beta\_2 \psi\_1'(1), \psi\_1(0) - 1 = \beta\_1 \psi\_1'(0), \\ \psi\_2(1) &= -\beta\_2 \psi\_2'(1), \psi\_2(0) - 1 = \beta\_1 \psi\_2'(0). \end{aligned} \tag{12}$$

where β1 = *<sup>H</sup>*1μ/*d*, β2 = *<sup>H</sup>*2μ/*d* are slip factors. Equations (7)–(9) and (12) will be solved by the HAM. The expression for pressure can be deduced from Equations (2)–(4) as follows:

$$-\frac{p}{\rho} = \frac{\mathcal{W}^2 \Lambda \mathbf{x}\_1^2}{2d} + \frac{1}{2} \phi\_3^2 - \gamma \phi\_{3, x\_3} + A \tag{13}$$

where Λ, *A* are constants and

$$
\Lambda = \left(\psi\_3^{\prime}\right)^2 - \psi \circ \psi\_3^{\prime \prime \prime} - \frac{1}{R} \psi\_3^{\prime \prime \prime \prime} = \left(\psi\_3^{\prime}(0)\right)^2 - \frac{1}{R} \psi\_3^{\prime \prime \prime \prime}(0). \tag{14}
$$

If 2*l* is the width of the slider with ambient pressure ρ0, then Equation (13) gives

$$p - p\_0 = -\rho \frac{\Lambda \mathcal{W}^2 \{\mathbf{x}\_1^2 - l^2\}}{2d^2}. \tag{15}$$

The relationship between depth and lift can be expressed as follows:

$$L = \int\_{-1}^{1} (p - p\_0) dx = \frac{2\rho W^2 l^3}{3d^2} \Lambda. \tag{16}$$

where <sup>2</sup>ρ*W*2*l*3/-3*d*<sup>2</sup>is normalized factor. The relationship between depth and drag in the *x*1− direction is

$$D\_{\mathbf{x}\_1} = -\int\_{-1}^{1} \mu \frac{\partial \phi\_1}{\partial \mathbf{x}\_3} \vert\_{z=d} d\mathbf{x}\_1 = -\frac{2\mu Ul}{d} \psi\_1'(1). \tag{17}$$

Similarly, <sup>2</sup>μ*Ul*/*d* is the normalized factor of drag in the *x*1− direction, which is −<sup>ψ</sup>/1 (1), while −<sup>ψ</sup>/2(1) is normalized drag for the *x*2− direction:

$$D\_{x2} = -\int\_{-1}^{1} \mu \frac{\partial \phi\_2}{\partial x\_3} \vert\_{z=d} d\mathbf{x}\_1 = -\frac{2\mu V l}{d} \psi\_2'(1). \tag{18}$$

Similarly, from Figure 1b, a circular slider can be seen, where *L* is the radius of the slider (which can be assumed to be comparatively bigger than the width). Since the slider is levitated, the axes on the slider can be fixed so that the ground is moving with a velocity component in the *x*1− direction. For the circular slider, a similar transform [18] helps to reduce the partial differential equations into ordinary differential equations:

$$\mathbf{x}\_1 = \mathcal{U}\boldsymbol{\psi}\_5(\boldsymbol{\varsigma}) + \frac{\mathcal{W}}{d} \mathbf{x}\_1 \boldsymbol{\psi}\_4^{\prime}(\boldsymbol{\varsigma}),\\\mathbf{x}\_2 = \frac{\mathcal{W}}{d} \mathbf{x}\_2 \boldsymbol{\psi}\_4^{\prime}(\boldsymbol{\varsigma}),\\\mathbf{x}\_3 = -2\mathcal{W}\boldsymbol{\psi}\_4(\boldsymbol{\varsigma}).\tag{19}$$

With the help of Equation (19), Equations (2)–(4) take the following form

$$
\psi\_4^{\dot{v}\upsilon} - 2R\psi\_4 \psi\_4^{\upsilon'/'} - M^2 \psi\_4^{\upsilon'} = 0 \tag{20}
$$

$$
\psi\_5^{\prime \prime} - R \Big( \psi\_5 \psi\_4^{\prime} - 2 \psi\_4 \psi\_5^{\prime} \Big) - M^2 \psi\_5 = 0 \tag{21}
$$

$$-\frac{p}{\rho} = \frac{\mathcal{W}^2 \Lambda(\mathbf{x}\_1^2 + \mathbf{x}\_2^2)}{2d} + \frac{1}{2}\mathbf{x}\_3^2 - \chi \mathbf{x}\_{3,x\_3} + \mathcal{C} \tag{22}$$

in which Λ, *C* are constants and

$$
\Lambda\_1 = \left(\psi\_4^{'}(0)\right)^2 - \frac{1}{R} \psi\_4^{'//} (0). \tag{23}
$$

The boundary conditions on *x*3 = 0&*d* :

$$\begin{aligned} \psi\_4'(0) &= \beta\_1 \psi\_4'^{'/}(0) = , \psi\_4(0) = 0, \\ \psi\_4(1) &= 1/2, \psi\_4'(1) = -\beta\_2 \psi\_4^{'/}(1), \\ \psi\_5(1) &= -\beta\_2 \psi\_5'(1), \psi\_5(0) - 1 = \beta\_1 \psi\_5'(0). \end{aligned} \tag{24}$$

To normalize the lift, integrating the bottom of the slider as a result of the normalized factor can be expressed as πρ*W*2*l*4/4*d*.

$$L = \frac{4d}{\pi \rho W^2 l^4} \iint\limits\_{\text{s}} (p - p\_0) ds = \frac{1}{R^3} \Lambda. \tag{25}$$

The relationship between depth and drag in the *x*1− direction is

$$D\_{\mathbf{x}\_1} = \frac{d}{\pi \mu l L l^2} \iint\_s H\_{\mathbf{x}\_3 \mathbf{x}\_1} ds = -\frac{1}{R^3} \psi\_5^{'}(1). \tag{26}$$

*Mathematics* **2019**, *7*, 748
